Wellbores are formed by rotating a drill bit carried at an end of an assembly commonly referred to as a bottom hole assembly or “BHA.” Herein, the BHA is used to mean the bottom hole assembly with or without the drill bit. The BHA is conveyed into the wellbore by a drill pipe or coiled-tubing. The rotation of the drill bit is effected by rotating the drill pipe and/or by a mud motor depending upon the tubing used. BHAs generally may include one or more formation evaluation sensors, such as sensors for measuring the resistivity, porosity, and density of the formation. Such BHAs may also include devices to determine the BHA inclination and azimuth, pressure sensors, vibration sensors, temperature sensors, gamma ray devices, and devices that aid in orienting the drill bit a particular direction and to change the drilling direction, Acoustic and resistivity devices may also be included for determining bed boundaries around and in some cases in front of the drill bit.
In practice, the BHAs are manufactured for specific applications and each such version usually contains only a selected number of devices and sensors. Additionally, BHAs may have limited data processing capabilities and do not compute the parameters downhole that can be used to control the drilling operations. Instead, BHAs may transmit data or partial answers uphole via a relatively small data-rate telemetry system. The drilling decisions are made at the surface based on the information provided by the BHA, data gathered during drilling of prior wellbores, and geophysical or seismic maps of the field. Drilling parameters, such as the weight-on-bit, drilling fluid flow rate, and drill bit RPM are usually measured and controlled at the surface.
The operating or useful life of the drill bit, mud motor, bearing assembly, and other elements of the BHA depends upon the manner in which such devices are operated and the down-hole conditions. This includes rock type and drilling conditions such as pressure, temperature, differential pressure across the mud motor, rotational speed, torque, vibration, drilling fluid flow rate, force on the drill bit or the weight-on-bit (“WOB”), type of the drilling fluid used, and the condition of the radial and axial bearings.
If any of the essential BHA components fails or, becomes relatively ineffective, the drilling operation must be shut down to pull out the drill string from the borehole to replace or repair such a component. Such premature failures can significantly increase the drilling cost Additionally, BHAs that contain multiple components must be designed to optimize drilling performance, especially with regards to directional drilling where the borehole is curved either using a steerable motor system, a point-the-bit rotary steerable system, or other type of directional drilling system. Parameters such as directional drilling ability and performance, location of stabilizers, BHA sag, as well as others are considered in the design of the placement of the various components of a BHA.
Modeling may also be used to optimize the design of the BHA, especially in directional drilling. BHA modeling enables many critical applications such as: (1) designing a BHA to optimize directional performance; (2) optimizing stabilizer locations to minimize vibration and increase down-hole tool reliability; and (3) improving survey data by correcting the BHA sag, etc. However, there are many challenges in developing a computationally efficient, flexible, and accurate BHA model, including: (1) nonlinear differential equations; (2) unknown upper boundary conditions at the tangential point (location and orientation); (3) unknown boundary conditions at stabilizers; (4) collars and wellbore wall contacts; and (5) large deformation caused by the bent housing motor or the bend in rotary steerable tools, etc.
FIG. 1 illustrates a directional well with a lateral borehole A drilled using a drillstring E with a drill bit at its distal end. The kick-off point B is the beginning of the build section C. A build section may be performed at a planned build-tip rate until the desired angle or end-of-build D is achieved. The build-up rate is normally expressed in terms of degrees-per-hundred-feet (deg/100′), which is simply the measured change in angle divided by the measured depth drilled. The build rate, or angle-changing capability of a motor or a rotary steerable system, depends on the extent to which the combination of bend and stabilizers and/or pads cause the bit to be offset from the center line of a straight borehole. Increased bit offset results in higher build rate. Increased bit offset, however, results in increased side loads when kicking off the borehole wall or when the motor is rotated in the borehole. High bit side loads can cause damage to the gage or bearings of the bit, and limit motor life by causing driveshaft fatigue, radial bearing wear, and stator damage. Stabilizer loads and associated wear also increase.
BHA modeling is used to analyze forces on the bit and stabilizers and bending stresses at connections and critical cross-section changes, with assemblies oriented in the model both highside and lowside. BHA modeling is also used in well planning for predicting the capabilities and tendencies of each BHA that is planned to be run. BHA modeling identifies the response of each BHA to variation in operating parameters such as weight on bit, overgage or undergage hole, stabilizer wear, and formation tendencies.
Various types of directional BHA models exist, as are well known by one skilled in the art. Two kinds of models are commonly used, the finite-element method and the semi-analytical method. Many BHA models using the finite-element analysis method are based on the small deformation theory. Thus, they have been shown to be not accurate enough to model steerable assemblies such as motor or rotary steerable systems. Finite-element modeling is also cumbersome in handling the collars and wellbore contact. To accurately model steerable systems, the semi-analytical methods are usually required; but semi-analytical methods may be inflexible. They are often designed to analyze some specific BHA models and are often limited to BHA with rather simple configurations.
The general solution, developed by Arthur Lubinski in the 1950's, to solve for the continuous beam-column model (fourth-order nonlinear differential equations) can be expressed in the form of:
                              x          ⁡                      (            z            )                          =                              P            1                    +                                    P              2                        ×                          cos              (                                                                    W                    EI                                                  ⁢                z                            )                                +                                    P              3                        ×                          sin              (                                                                    W                    EI                                                  ⁢                z                            )                                +                                                    H                ′                            W                        ⁢            z                    +                                                    q                ⁢                                                                  ⁢                sin                ⁢                                                                  ⁢                α                                            2                ⁢                W                                      ×                          z              2                                                          (        1        )            In which P1, P2, and P3 are three variables H′ is the “normalized” side force acting on the left hand side of the segment, W is the weight-on-bit; q is the unit weight of the drill string and α is the inclination angle. Thus, for each segment in the BHA model, there are four unknowns: P1, P2, P3 and H′. Note P1, P2, and P3 are called the segment profiles in this paper.
A BHA model usually consists of components such as a drill hit, stabilizers, cross-overs, bends, and offset pads, etc. For a BHA with N components, there will be (N+1) sections in the model. The last section in the model is bounded by the top component and the tangential point where the collars are in contact with the wellbore wall hi each section, there will be four unknowns as solved by using Lubinski's BHA equations. Thus, for an N-component BHA, there are “approximately” 4×(N+1) unknowns associated with 4×(N+1) non-linear equations in the model.
Deriving the 4×(N+1) non-linear equations or even iterating 4×(N+1) unknowns is extremely difficult, especially when N becomes large. For example, a typical BHA with 2-3 stabilizers, 3-4 cross-over, and one bend will have an N of at least 6. Thus, there are approximately 28 non-linear equations to be derived and 28 unknowns to be solved. There are two main challenges to be overcome when using the closed-form solutions such as Lubinski's equations: (1) to find a process to construct the model dynamically; and (2) to reduce the unknowns to a manageable number for iteration.